### Models of expansions of $\Bbb N$ with no end extensions

by Shelah. [Sh:937]

Math Logic Quarterly, 2011

We deal with models of Peano arithmetic (specifically with
a question of Ali Enayat).
The methods are from creature forcing.
We find an expansion of N such that its theory has
models with no (elementary) end extensions. In fact there is a Borel
uncountable set of subsets of N
such that expanding N by any uncountably many of them
suffice. Also we find arithmetically closed A with
no definably closed ultrafilter on it

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